A hat company is designing a one size fits all hat with a strap in the back that makes the hat smaller or larger. head sizes normally range from 51 to 64 centimeters. what absolute value inequality models the different sizes of the hat? graph the solution
the middle value is (51+64)/2 = 57.5
That is 6.5 away from each end, so
|x-57.5| <= 6.5
See the graph at
http://www.wolframalpha.com/input/?i=|x-57.5|+%3C%3D+6.5
The absolute value inequality that models the different sizes of the hat is:
| x - 57.5 | ≤ 6.5
Where x represents the head size in centimeters and 57.5 represents the average head size. The inequality states that the difference between the head size and the average head size must be less than or equal to 6.5 centimeters.
To graph the solution, you can create a number line with the range of head sizes from 51 to 64. Then, mark the average head size (57.5) and draw a solid line to represent all the values that satisfy the inequality.
Here's a visual representation of the number line:
______________________________
51 57.5 64
The solid line should be drawn between the numbers 51 and 64.
To represent the different sizes of the hat using an absolute value inequality, we need to consider the variation from the average head size of 57.5 centimeters. The absolute value inequality will capture values both above and below this average.
Let's denote the average head size as "a" and the adjustable range as "r." The absolute value inequality can be written as:
| x - a | ≤ r
In this case, x represents the head size and a represents the average head size. The adjustable range, r, determines how much variation is allowed.
Based on the information given, the average head size (a) is 57.5 centimeters, and the adjustable range (r) is 6.5 centimeters (the difference between the smallest and largest head sizes). Therefore, the absolute value inequality is:
| x - 57.5 | ≤ 6.5
To graph the solution on a number line, we need to determine the values of x that satisfy this inequality.
We can break down the absolute value inequality into two separate inequalities:
1. x - 57.5 ≤ 6.5
2. -(x - 57.5) ≤ 6.5
Solving each inequality:
1. x ≤ 6.5 + 57.5
x ≤ 64
2. -x + 57.5 ≤ 6.5
x ≥ 51
Combining the two inequalities, we get:
51 ≤ x ≤ 64
This means that any head size between 51 and 64 centimeters (inclusive) will fit the one size fits all hat.
To graph the solution, draw a number line and mark the values from 51 to 64. You can either use open circles or closed circles to indicate inclusion or exclusion of the endpoints.